Introduction to Mixed Modelling (eBook)

Beyond Regression and Analysis of Variance

N. W. Galwey (Autor)

eBook Download: EPUB

2014 | 2. Auflage
John Wiley & Sons (Verlag)
978-1-118-86182-0 (ISBN)

Infos

This book first introduces the criterion of REstricted Maximum Likelihood (REML) for the fitting of a mixed model to data before illustrating how to apply mixed model analysis to a wide range of situations, how to estimate the variance due to each random-effect term in the model, and how to obtain and interpret Best Linear Unbiased Predictors (BLUPs) estimates of individual effects that take account of their random nature.  

It is intended to be an introductory guide to a relatively advanced specialised topic, and to convince the reader that mixed modelling is neither so specialised nor so difficult as it may at first appear.

 

This edition presents new material in the following areas:

• Use of mixed models for meta-analysis of a set of experiments, especially clinical trials.
• The Bayesian interpretation of best linear unbiased predictors (BLUPs).
• The multiple-testing problem and the shrinkage of BLUPs as a defence against the ‘Winner’s Curse’.
• The implementation of mixed models in the statistical software SAS.
• Increasing the precision of significance tests in mixed models, by estimation of the denominator degrees of freedom (the Kenward-Roger method).
• Tests for comparison of non-nested mixed models, using the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC).

---

Mixed modelling is very useful, and easier than you think! Mixed modelling is now well established as a powerful approach to statistical data analysis. It is based on the recognition of random-effect terms in statistical models, leading to inferences and estimates that have much wider applicability and are more realistic than those otherwise obtained. Introduction to Mixed Modelling leads the reader into mixed modelling as a natural extension of two more familiar methods, regression analysis and analysis of variance. It provides practical guidance combined with a clear explanation of the underlying concepts. Like the first edition, this new edition shows diverse applications of mixed models, provides guidance on the identification of random-effect terms, and explains how to obtain and interpret best linear unbiased predictors (BLUPs). It also introduces several important new topics, including the following: Use of the software SAS, in addition to GenStat and R. Meta-analysis and the multiple testing problem. The Bayesian interpretation of mixed models. Including numerous practical exercises with solutions, this book provides an ideal introduction to mixed modelling for final year undergraduate students, postgraduate students and professional researchers. It will appeal to readers from a wide range of scientific disciplines including statistics, biology, bioinformatics, medicine, agriculture, engineering, economics, archaeology and geography. Praise for the first edition: One of the main strengths of the text is the bridge it provides between traditional analysis of variance and regression models and the more recently developed class of mixed models...Each chapter is well-motivated by at least one carefully chosen example...demonstrating the broad applicability of mixed models in many different disciplines...most readers will likely learn something new, and those previously unfamiliar with mixed models will obtain a solid foundation on this topic. Kerrie Nelson University of South Carolina, in American Statistician, 2007

Autor

Nicholas W. Galwey, Statistical Consultant, GlaxoSmithKline, Harlow, Essex, UK.

Inhalt

Preface

1. The need for more than one random-effect term when fitting a regression line

2. The need for more than one random-effect term in a designed experiment

3. Estimation of the variances of random-effect terms

4. Interval estimates for fixed-effect terms in mixed models

5. Estimation of random effects in mixed models: Best Linear Unbiased Predictors (BLUPs)

6. More advanced mixed models for more elaborate data sets

7. Three case studies

8. Meta-analysis and the multiple testing problem

9. The use of mixed models for the analysis of unbalanced experimental designs

10. Beyond mixed modeling

11. Why is the criterion for fitting mixed models called REsidual Maximum Likelihood?

Index

Textauszug

Chapter 1
The need for more than one random-effect term when fitting a regression line

1.1 A data set with several observations of variable Y at each value of variable X

One of the commonest, and simplest, uses of statistical analysis is the fitting of a straight line, known for historical reasons as a regression line, to describe the relationship between an explanatory variable, X and a response variable, Y. The departure of the values of Y from this line is called the residual variation, and is regarded as random. It is natural to ask whether the part of the variation in Y that is explained by the relationship with X is more than could reasonably be expected by chance: or more formally, whether it is significant relative to the residual variation. This is a simple regression analysis, and for many data sets it is all that is required. However, in some cases, several observations of Y are taken at each value of X. The data then form natural groups, and it may no longer be appropriate to analyse them as though every observation were independent: observations of Y at the same value of X may lie at a similar distance from the line. We may then be able to recognize two sources of random variation, namely

• variation among groups
• variation among observations within each group.

This is one of the simplest situations in which it is necessary to consider the possibility that there may be more than a single stratum of random variation—or, in the language of mixed modelling, that a model with more than one random-effect term may be required. In this chapter, we will examine a data set of this type and explore how the usual regression analysis is modified by the fact that the data form natural groups.

We will explore this question in a data set that relates the prices of houses in England to their latitude. There is no doubt that houses cost more in the south of England than in the north: these data will not lead to any new conclusions, but they will illustrate this trend, and the methods used to explore it. The data are displayed in a spreadsheet in Table 1.1. The first cell in each column contains the name of the variable held in that column. The variables ‘latitude’ and ‘price_pounds’ are variates—lists of observations that can take any numerical value, the commonest kind of data for statistical analysis. However, the observations of the variable ‘town’ can take only certain permitted values—in this case, the names of the 11 towns under consideration. A variable of this type is called a factor, and the exclamation mark (!) after its name indicates that ‘town’ is a factor. The towns are the groups of observations: within each town, all the houses are at nearly the same latitude, and the latitude of the town is represented by a single value in this data set. In contrast, the price of each house is potentially unique. The conventions introduced here apply to all other spreadsheets displayed in this book.

Table 1.1 Prices of houses in a sample of English towns and their latitudes.

A

B

C

A

B

C

1

town!

latitude

price_pounds

34

Crewe

53.0998

84950

2

Bradford

53.7947

39950

35

Crewe

53.0998

112500

3

Bradford

53.7947

59950

36

Crewe

53.0998

140000

4

Bradford

53.7947

79950

37

Durham

54.7762

127950

5

Bradford

53.7947

79995

38

Durham

54.7762

157000

6

Bradford

53.7947

82500

39

Durham

54.7762

169950

7

Bradford

53.7947

105000

40

Newbury

51.4037

172950

8

Bradford

53.7947

125000

41

Newbury

51.4037

185000

9

Bradford

53.7947

139950

42

Newbury

51.4037

189995

10

Bradford

53.7947

145000

43

Newbury

51.4037

195000

11

Buxton

53.2591

120000

44

Newbury

51.4037

295000

12

Buxton

53.2591

139950

45

Newbury

51.4037

375000

13

Buxton

53.2591

149950

46

Newbury

51.4037

400000

14

Buxton

53.2591

154950

47

Newbury

51.4037

475000

15

Buxton

53.2591

159950

48

Ripon

54.1356

140000

16

Buxton

53.2591

159950

49

Ripon

54.1356

152000

17

Buxton

53.2591

175950

50

Ripon

54.1356

187950

18

Buxton

53.2591

399950

51

Ripon

54.1356

210000

19

Carlisle

54.8923

85000

52

Royal Leamington Spa

52.2876

147000

20

Carlisle

54.8923

89950

53

Royal Leamington Spa

52.2876

159950

21

Carlisle

54.8923

90000

54

Royal Leamington Spa

52.2876

182500

22

Carlisle

54.8923

103000

55

Royal Leamington Spa

52.2876

199950

23

Carlisle

54.8923

124950

56

Stoke-On-Trent

53.0041

69950

24

Carlisle

54.8923

128500

57

Stoke-On-Trent

53.0041

69950

25

Carlisle

54.8923

132500

58

Stoke-On-Trent

53.0041

75950

26

Carlisle

54.8923

135000

59

Stoke-On-Trent

53.0041

77500

27

Carlisle

54.8923

155000

60

Stoke-On-Trent

53.0041

87950

28

Carlisle

54.8923

158000

61

Stoke-On-Trent

53.0041

92000

29

Carlisle

54.8923

175000

62

Stoke-On-Trent

53.0041

94950

30

Chichester

50.8377

199950

63

Witney

51.7871

179950

31

Chichester

50.8377

299250

64

Witney

51.7871

189950

32

Chichester

50.8377

350000

65

Witney

51.7871

220000

33

Crewe

53.0998

77500

Source: Data obtained from an estate agent's website in October 2004.

Before commencing a formal analysis of this data set, we should note its limitations. A thorough investigation of the relationship between latitude and house price would take into account many factors besides those recorded here—the number of bedrooms in each house, its state of repair and improvement, other observable indicators of the desirability of its location, and so on. To some extent such sources of variation have been eliminated from the present sample by the choice of houses that are broadly similar: they are all ‘ordinary’ houses (no flats, maisonettes, etc.) and all have three, four or five bedrooms. The remaining sources of variation in price will contribute to the residual variation among houses in each town, and will be treated accordingly. We should also consider in what sense we can think of the latitude of houses as ‘explaining’ the variation in their prices. The easily measurable variable ‘latitude’ is...

Produkt-Details

• Erscheint lt. Verlag: 26.8.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Schlagworte: Ökologie / Methoden, Statistik • Angew. Wahrscheinlichkeitsrechn. u. Statistik / Modelle • Applicability • Applied Probability & Statistics - Models • Biowissenschaften • Data • established • Estimates • Experimental Design • extension • familiar • inferences • Leads • Life Sciences • Methods • Methods & Statistics in Ecology • Modelling • Ökologie / Methoden, Statistik • Otherwise • powerful approach • randomeffect • Reader • Realistic • Recognition • Statistical • Statistical Models • Statistics • Statistik • Terms • useful • Versuchsplanung • wider
• ISBN-10: 1-118-86182-5 / 1118861825
• ISBN-13: 978-1-118-86182-0 / 9781118861820